Geometry of Conformal and Quasiconformal Mappings

摘要：项目负责人：Christopher Bishop将研究共形和拟共形映射的几何性质，重点研究共形结构、双曲几何、低维拓扑和数值分析之间的相互作用。例如，摩尔定理指出，如果我们在2球上取一定的偏移集合，并将它们拓扑坍缩为点，那么我们得到一个新的拓扑球。PI将研究商映射何时可以是共形的，并将从这个一般的角度考虑一些具体的问题，包括共形焊接，约翰域的表征，Koebe猜想，Kleinian群的构造和其他动态对象。PI将继续他早期的工作，研究克莱因极限集的几何以及当我们变形极限集时维度的行为。PI还将继续他在计算几何、双曲几何和保角映射之间的联系方面的工作，并寻求快速计算黎曼映射和严格误差估计的新算法。特别是，他将研究使用中轴线（计算几何中的一个对象）计算共形映射，中轴线通过凸体与三维双曲几何紧密相连。保角映射在许多数学问题（复杂分析，动力系统，…）和各种应用（流体流动，脑映射，统计物理，微分方程的数值分析，…）中发挥着重要作用，因此我们必须对这些映射有很好的理论理解和在实践中计算它们的好方法。该提案通过研究与数学和计算机科学的其他部分（点集拓扑，三维双曲几何，Voronoi图）的新联系，加深了我们对共形映射的理论理解，并寻求利用这些联系来发明计算共形映射的新算法。例如，中轴线在计算机科学中是一个被广泛研究的对象（它是对物体形状的描述，在模式识别、机器人运动、生物学等领域有许多应用）。PI发现它也可以用来给出一个粗略但快速的共形映射近似值。PI将寻求改进这种方法，为保角映射提供更好的算法，并更好地理解将影响其其他应用的中轴线（例如，当被描述的对象仅发生一点点变化时，中轴线可能会急剧变化；这是一个严重的计算问题，可以通过将中轴线视为双曲几何中的对象而不是通常的欧几里得几何来解决）。中间轴和共形映射之间的联系也可能导致研究三维应用的新思路（其中不存在共形映射，但基于中间轴的映射仍然存在）。三维问题是最重要的应用，但我们的理解远远落后于二维的情况下，所以新的二维思想推广到更高的维度是很重要的。

AbstractAward: DMS-0405578Principal Investigator: Christopher BishopThe PI, Christopher Bishop, will study the geometric propertiesof conformal and quasiconformal mappings, focusing on theinteractions between conformal structures, hyperbolic geometry,low dimensional topology and numerical analysis. For example,Moore's theorem states that if we take a certain collection ofsets on the 2-sphere and topologically collapse them to pointsthen we obtain a new topological sphere. The PI will investigatewhen the quotient map can be conformal and will consider a numberof concrete problems from this general perspective, includingconformal welding, characterizations of John domains, Koebe'sconjecture, construction of Kleinian groups and other dynamicalobjects. The PI will continue his earlier work on the geometryof Kleinian limit sets and the behavior of the dimension as wedeform the limit set. The PI will also continue his work on theconnections between computational geometry, hyperbolic geometryand conformal mappings, and seek new algorithms which compute theRiemann mapping quickly and with rigorous error estimates. Inparticular he will investigate computing conformal maps using themedial axis (an object from computational geometry) which isclosely linked to 3-dimensional hyperbolic geometry via convexhulls.Conformal mappings are important both for their central role innumerous mathematical problems (complex analysis, dynamicalsystems,...) and in various applications (fluid flow, brainmapping, statistical physics, numerical analysis of differentialequations,...), so we must have a good theoretical understandingof these maps and good methods for computing them inpractice. The proposal deepens our theoretical understanding ofconformal maps by investigating new connections with other partsof mathematics and computer science (point set topology,3-dimensional hyperbolic geometry, Voronoi diagrams) and seeks touse these connections to invent new algorithms for computingconformal maps. For example, the medial axis is a widely studiedobject in computer science (it is a description of the shape onan object which has numerous applications in pattern recognition,robotic motion, biology,...). The PI discovered it can also beused to give a rough but fast approximation to conformal maps.The PI will seek to improve this method, giving better algorithmsfor conformal maps and also developing a better understanding ofthe medial axis which will impact its other applications (forexample, the medial axis can change drastically when the objectbeing described changes only a little; this is a seriouscomputational problem which can be addressed by thinking of themedial axis as an object in hyperbolic geometry instead of theusual Euclidean geometry). The connection between the medial axisand conformal maps may also lead to new ideas for studyingapplications in three dimensions (where conformal mappings do notexist, but maps based on the medial axis still do). Threedimensional problems are the most important for applications, butour understanding lags far behind the two dimensional case, sonew two dimensional ideas which generalize to higher dimensionsare important.